THE Yarrow Stalk Method and the probability distribution of yao

We have studied the I Ching hexagram based on the knowledge of binary arithmetic in the past issues, we will shed more light on the question of the relationship between the I Ching system and binary arithmetic next time.

Today, we’ll focus our discussion on how people make use of gua in I Ching to gain confidence in predicting future and attempt to control uncertainty in their own fate. Therefore, the Book of Changes is also interpreted as Book of Oracle or The Manual of Divination. So let’s begin with the divination.

Divination in Chinese history of past 3,000 years entails from a method using yarrow stalks (see Picture 1), aiming to risk control. The yarrow stalk method (大衍筮法）was popular in Western Zhou (c.11th century-770BC). The divination, consequently, became a primary method for prediction. The method can generate four numbers, among them, 6 and 9 are alterable, 8 and 7 are unalterable. Specifically, 6 stands for a changing yin yao, 8 stands for a resting yin yao; while 9 and 7 stand for changing and resting yang yao, respectively.

The word ‘changing’ here means the yao to be changed into its opposite, e.g., 6 to be changed to 7, and 9 to 8. The word ‘resting’ on the other hand means that 7 and 8 can no longer change.

The process of I Ching divination is essentially a random number generator. But what is the chances to obtain number 6, 7, 8 and 9 respectively?

Governed by the yarrow stalk method, the chances to obtain 6 is 1/16, to obtain 8 is 7/16, to obtain 9 and 7 are 3/16 and 5/16, respectively. In other words, the probability distribution of the yarrow stalk divination of changing yin yao: resting yang yao: resting yin yao: changing yang yao (6:7:8:9) is 1:5:7:3.

You may find that the chances for generating a yin yao (changing 1/16 + resting 7/16) and a yang yao (changing 3/16 + resting 5/16) are equal, both of 50 percent of the chance.

Equipped with this knowledge, you may also discover simplified replacements of the yarrow stalk method, numerous of them in daily use, are likely misleading. For example, some coin-flipping methods would generate the odds of four different kinds of yao as 1/4:1/4:1/4:1/4; some dice-tossing methods would generate the odds as 1/8:3/8:3/8:1/8, deviated way from the correct results. These replacing methods generate confusion rather than simplification.

However, the attempt for a simple method of divination is well grounded. Because the yarrow stalk method is extremely tedious. As a solemn and complicated ceremony, it demands trained professional to conduct this intricate process.

Naturally, the yarrow stalk method lost its aura and attractiveness to ordinary diviners over time. Zhu Xi (1130-1200), a renowned thinker and key master of Confucianism, dug deep into the method and revealed that the resting yao are much more frequent than changing yao.

Why, then, is the yarrow stalk method important to abide by? The text of the classic and the mainstream interpretation of Zhou Yi as reflected in the books of “Ten Wings” are all based on the yao and gua divined through the yarrow stalk method. Although the terms and words in the text are largely descriptive, the divination is prescriptive. With wrong ways and incorrect probabilities the divined yao and gua deem the text inapplicable.

Then how to make ends meet? I designed a simple scheme to assure the probability pattern exactly the same to the yarrow stalk method. Since there is no such thing of a regular 12-faced polyhedron, we have no way to have a simple dice for the purpose. My recommendation instead is to toss a pair of fair coins twice for one yao. Repeat three times to get a 3-yao gua, or six times to get a 6-yao gua. The simple process of this random number generator is covered in Picture 2.

Distribution of the probability of I Ching gua

A 6-yao gua is composed of two 3-yao gua. Thus, we can get a top 3-yao gua through first three tosses. Another three tosses generate a bottom 3-yao gua. A 6-yao gua will be formed accordingly. The method helps us get a 6-yao gua easily and also indicates whether the yao is changeable or unchangeable..

How should we calculate the probability of each I Ching gua then? In Kun Gua, the hexagram of pure yin yao, each yao can be either number 6 or 8, which means Kun Gua has 64 (26) variations. If all the six yin yao are resting, for example 888888 (the numbers from right to left represents bottom yao to top yao), its probability is (7/16) 6. The probability of Kun Gua with six changing yin yao, for example 666666, is (1/16) 6. The former probability is 76 = 117,649 times of the latter’s. Similarly, if a pure yang Qian Gua has only one changing yang yao, for example 977777, its probability is (5/16)5*(3/16), which is 12.1 times of the probability of a Qian Gua with four changing yang yao (799997), which is (5/16)2*(3/16)4.

The generation of each yao is independent. Thus, the position of yao doesn’t play a role here. For example, the probability of number 8 is 7/16 no matter where the yao is positioned. The same rule applies to number 9. Any gua’s probability, therefore, depends on the combination of the number of yin yao and yang yao as well as whether the yao is resting or changing. In short, it depends on the combination of the four numbers, which is calculated using the method of combination and not permutation.

Actually, the 64 variations of Kun Gua have only seven unique values, for example (1/16)m*(7/16)n. Both “m” and “n” are natural numbers ranging from 0 to 6, sum to 6. Thus, the probability of Kun Gua is 0.015625. So is Qian Gua. However, the probability of generating number 9 for a changing yang yao is 3/16, while that of number 7 for a resting yang yao is 5/16.

1.5625 percent, i.e. 1/64, means the 64 gua share the equal chance to be divined. Each gua has 64 variations, totaling 4,096 (642) variations. But the number of unique values of probability is 84, much smaller. Please note that the combination of yin yao and yang yao, divined either changing or resting, regardless where is the yao positioned.

Alternatively, we can calculate the probability of eight 3-yao gua. Multiply the probability of top 3-yao gua (上卦）and the probability of bottom 3-yao gua (下卦) we can easily get the probability of all 64 6-yao gua, as seen in Picture 3. Bagua are of four groups. Group A, zero-yang gua, has Earth only; Group B, three-yang gua, has Sky only; Group C, one-yang gua, has three gua: Thunder, Water and Mountain; Group D, two-yang gua, has another three gua: Valley, Fire and Wind. The probability of each gua is 1/8, or 12.5 percent. There are three gua in either Group C or Group D. So their probabilities are both 37.5 percent.

Let’s illustrate this conclusion with Tai Gua (top Earth-bottom Sky). Two Tai Gua are generated through divination, for example, A (688997) and B (868799). The top gua in A is Earth (688) and that in B is also Earth (868). Both Earths have one changing yin yao and two resting yin yang, so both have a probability of 0.011962891. The bottom gua in A is Sky (997) and that in B is also Sky (799). As of two changing yang yao and one resting yang yao, both have a probability of 0.010986328. We multiply these two probabilities, resulting 0.000131428. We thus know that the probability of either Tai Gua A or Tai Gua B are identical 0.01314 percent.

In practical applications, there is no need to have to be that precise. Simply, the probability of each of the 64 gua is 1.56%. There are six groups of the 64 gua: zero-yang (1 gua), one-yang (6 gua), two-yang (15 gua), three-yang (20 gua), four-yang (15 gua), five-yang (6 gua), and six-yang (1 gua). Their probabilities are different. In percentage they are 1.56, 9.36, 23.4, 31.36, 23.4, 9.36, and 1.56, respectively.